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      SUBROUTINE <a name="DLANV2.1"></a><a href="dlanv2.f.html#DLANV2.1">DLANV2</a>( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      DOUBLE PRECISION   A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DLANV2.14"></a><a href="dlanv2.f.html#DLANV2.1">DLANV2</a> computes the Schur factorization of a real 2-by-2 nonsymmetric
</span><span class="comment">*</span><span class="comment">  matrix in standard form:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">       [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
</span><span class="comment">*</span><span class="comment">       [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where either
</span><span class="comment">*</span><span class="comment">  1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
</span><span class="comment">*</span><span class="comment">  2) AA = DD and BB*CC &lt; 0, so that AA + or - sqrt(BB*CC) are complex
</span><span class="comment">*</span><span class="comment">  conjugate eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  B       (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  C       (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  D       (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          On entry, the elements of the input matrix.
</span><span class="comment">*</span><span class="comment">          On exit, they are overwritten by the elements of the
</span><span class="comment">*</span><span class="comment">          standardised Schur form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT1R    (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  RT1I    (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  RT2R    (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  RT2I    (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The real and imaginary parts of the eigenvalues. If the
</span><span class="comment">*</span><span class="comment">          eigenvalues are a complex conjugate pair, RT1I &gt; 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CS      (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  SN      (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          Parameters of the rotation matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Modified by V. Sima, Research Institute for Informatics, Bucharest,
</span><span class="comment">*</span><span class="comment">  Romania, to reduce the risk of cancellation errors,
</span><span class="comment">*</span><span class="comment">  when computing real eigenvalues, and to ensure, if possible, that
</span><span class="comment">*</span><span class="comment">  abs(RT1R) &gt;= abs(RT2R).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      DOUBLE PRECISION   ZERO, HALF, ONE
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   MULTPL
      PARAMETER          ( MULTPL = 4.0D+0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      DOUBLE PRECISION   AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
     $                   SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      DOUBLE PRECISION   <a name="DLAMCH.68"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, <a name="DLAPY2.68"></a><a href="dlapy2.f.html#DLAPY2.1">DLAPY2</a>
      EXTERNAL           <a name="DLAMCH.69"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, <a name="DLAPY2.69"></a><a href="dlapy2.f.html#DLAPY2.1">DLAPY2</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span>      EPS = <a name="DLAMCH.76"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'P'</span> )
      IF( C.EQ.ZERO ) THEN
         CS = ONE
         SN = ZERO
         GO TO 10
<span class="comment">*</span><span class="comment">
</span>      ELSE IF( B.EQ.ZERO ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Swap rows and columns
</span><span class="comment">*</span><span class="comment">
</span>         CS = ZERO
         SN = ONE
         TEMP = D
         D = A
         A = TEMP
         B = -C
         C = ZERO
         GO TO 10
      ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
     $          THEN
         CS = ONE
         SN = ZERO
         GO TO 10
      ELSE
<span class="comment">*</span><span class="comment">
</span>         TEMP = A - D
         P = HALF*TEMP
         BCMAX = MAX( ABS( B ), ABS( C ) )
         BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
         SCALE = MAX( ABS( P ), BCMAX )
         Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        If Z is of the order of the machine accuracy, postpone the
</span><span class="comment">*</span><span class="comment">        decision on the nature of eigenvalues
</span><span class="comment">*</span><span class="comment">
</span>         IF( Z.GE.MULTPL*EPS ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Real eigenvalues. Compute A and D.
</span><span class="comment">*</span><span class="comment">
</span>            Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
            A = D + Z
            D = D - ( BCMAX / Z )*BCMIS
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute B and the rotation matrix
</span><span class="comment">*</span><span class="comment">
</span>            TAU = <a name="DLAPY2.121"></a><a href="dlapy2.f.html#DLAPY2.1">DLAPY2</a>( C, Z )
            CS = Z / TAU
            SN = C / TAU
            B = B - C
            C = ZERO
         ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Complex eigenvalues, or real (almost) equal eigenvalues.
</span><span class="comment">*</span><span class="comment">           Make diagonal elements equal.
</span><span class="comment">*</span><span class="comment">
</span>            SIGMA = B + C
            TAU = <a name="DLAPY2.132"></a><a href="dlapy2.f.html#DLAPY2.1">DLAPY2</a>( SIGMA, TEMP )
            CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
            SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute [ AA  BB ] = [ A  B ] [ CS -SN ]
</span><span class="comment">*</span><span class="comment">                   [ CC  DD ]   [ C  D ] [ SN  CS ]
</span><span class="comment">*</span><span class="comment">
</span>            AA = A*CS + B*SN
            BB = -A*SN + B*CS
            CC = C*CS + D*SN
            DD = -C*SN + D*CS
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute [ A  B ] = [ CS  SN ] [ AA  BB ]
</span><span class="comment">*</span><span class="comment">                   [ C  D ]   [-SN  CS ] [ CC  DD ]
</span><span class="comment">*</span><span class="comment">
</span>            A = AA*CS + CC*SN
            B = BB*CS + DD*SN
            C = -AA*SN + CC*CS
            D = -BB*SN + DD*CS
<span class="comment">*</span><span class="comment">
</span>            TEMP = HALF*( A+D )
            A = TEMP
            D = TEMP
<span class="comment">*</span><span class="comment">
</span>            IF( C.NE.ZERO ) THEN
               IF( B.NE.ZERO ) THEN
                  IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                    Real eigenvalues: reduce to upper triangular form
</span><span class="comment">*</span><span class="comment">
</span>                     SAB = SQRT( ABS( B ) )
                     SAC = SQRT( ABS( C ) )
                     P = SIGN( SAB*SAC, C )
                     TAU = ONE / SQRT( ABS( B+C ) )
                     A = TEMP + P
                     D = TEMP - P
                     B = B - C
                     C = ZERO
                     CS1 = SAB*TAU
                     SN1 = SAC*TAU
                     TEMP = CS*CS1 - SN*SN1
                     SN = CS*SN1 + SN*CS1
                     CS = TEMP
                  END IF
               ELSE
                  B = -C
                  C = ZERO
                  TEMP = CS
                  CS = -SN
                  SN = TEMP
               END IF
            END IF
         END IF
<span class="comment">*</span><span class="comment">
</span>      END IF
<span class="comment">*</span><span class="comment">
</span>   10 CONTINUE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
</span><span class="comment">*</span><span class="comment">
</span>      RT1R = A
      RT2R = D
      IF( C.EQ.ZERO ) THEN
         RT1I = ZERO
         RT2I = ZERO
      ELSE
         RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
         RT2I = -RT1I
      END IF
      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="DLANV2.203"></a><a href="dlanv2.f.html#DLANV2.1">DLANV2</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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